symmetry relations should be of the same variance type ie both upper or both

# Symmetry relations should be of the same variance

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symmetry relations should be of the same variance type , i.e. both upper or both lower. Another important remark is that the symmetry and anti-symmetry characteristic of a ten- sor is invariant under coordinate transformations. Hence, a symmetric/anti-symmetric tensor in one coordinate system is symmetric/anti-symmetric in all other coordinate sys- tems. Similarly, a tensor which is neither symmetric nor anti-symmetric in one coordinate system remains so in all other coordinate systems. [40] Finally, for a symmetric tensor A ij and an anti-symmetric tensor B ij (or the other way around) we have the following useful and widely used identity: A ij B ij = 0 (97) This is because an exchange of indices will change the sign of one tensor only and this will change the sign of the term in the summation resulting in having a sum of terms which is identically zero due to the fact that each term in the sum has its own negation. 2.7 Exercises 2.1 Make a sketch of a rank-2 tensor A ij in a 4D space similar to Figure 11. What this tensor looks like? 2.2 What are the two main types of notation used for labeling tensors? State two names for each. 2.3 Make a detailed comparison between the two types of notation in the previous question stating any advantages or disadvantages in using one of these notations or the other. [40] In this context, like many other contexts in this book, there are certain restrictions on the type and conditions of the coordinate transformations under which such statements are valid. However, these details cannot be discussed here due to the elementary level of this book.
2.7 Exercises 74 In which context each one of these notations is more appropriate to use than the other? 2.4 What is the principle of invariance of tensors and why it is one of the main reasons for the use of tensors in science? 2.5 What are the two different meanings of the term “covariant” in tensor calculus? 2.6 State the type of each one of the following tensors considering the number and position of indices (i.e. covariant, contravariant, rank, scalar, vector, etc.): a i B jk i f b k C ji 2.7 Define the following technical terms which are related to tensors: term, expression, equality, order, rank, zero tensor, unit tensor, free index, dummy index, covariant, contravariant, and mixed. 2.8 Which of the following is a scalar, vector or rank-2 tensor: temperature, stress, cross product of two vectors, dot product of two vectors, and rate of strain? 2.9 What is the number of entries of a rank-0 tensor in a 2D space and in a 5D space? What is the number of entries of a rank-1 tensor in these spaces? 2.10 What is the difference between the order and rank of a tensor considering the different conventions in this regard? 2.11 What is the number of entries of a rank-3 tensor in a 4D space? What is the number of entries of a rank-4 tensor in a 3D space?

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• Summer '20
• Rajendra Paramanik
• Tensor, Coordinate system, Polar coordinate system, Coordinate systems

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